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A Quantitative Selberg's Lemma

Tsachik Gelander, Raz Slutsky

Abstract

We show that an arithmetic lattice $Γ$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $δ(v)$ where $v = μ(G/Γ)$ is the co-volume of the lattice. We prove that $δ$ is polynomial in general and poly-logarithmic under GRH. We then show that this poly-logarithmic bound is almost optimal, by constructing certain lattices with torsion elements of order $\sim \frac{\log v}{\log \log v}$.

A Quantitative Selberg's Lemma

Abstract

We show that an arithmetic lattice in a semi-simple Lie group contains a torsion-free subgroup of index where is the co-volume of the lattice. We prove that is polynomial in general and poly-logarithmic under GRH. We then show that this poly-logarithmic bound is almost optimal, by constructing certain lattices with torsion elements of order .
Paper Structure (5 sections, 10 theorems, 7 equations)

This paper contains 5 sections, 10 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Upper Bound
  3. Lower Bound
  4. Size of Finite subgroups
  5. Possible application

Key Result

Theorem 1.1

A finitely generated group $\Gamma \leq GL_n(k)$ where $k$ is a field of characteristic zero has a torsion-free subgroup of finite index.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2: GRH
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: GRH
  • proof
  • proof : Proof of Thm. \ref{['thm: intro GRH main theorem']}
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 7 more